Yield to Maturity..Continued

Transcription

1 LECTURE 3 Hamza Ali Malik Econ 315: Mony and Banking Wintr 006 Yild to Maturity..Continud 3) - Coupon Bonds is a dbt instrumnt in which fixd intrst paymnts ar mad throughout th lif of th contract but th principal is rpaid in lump sum at th nd of th trm. $C $C $C $FV $PV L 1 i ( 1 i) ( 1 i) n Exampl: A bond that promiss to pay intrst of /yar for 5 yars, so that th coupon rat is 10% (/). Th PV is th following paymnt stram: $PV ( ) ( ) 3 ( ) 4 ( ) 5 ( ) 5 $ 5 ( ) Notic that th PV of th bond is th sam as its fac valu. Coupon rat and yild ar idntical. If a particular bond trads at lss than its fac valu, it is trading at a discount. On that trads at mor than its fac valu is bing trad at a prmium. 1

2 Thr intrsting facts: 1) Whn th coupon bond is pricd at its fac valu, th yild to maturity quals th coupon rat. ) Th pric of a coupon bond and th yild to maturity ar ngativly rlatd. 3) Th yild to maturity is gratr than th coupon rat whn th bond pric is blow its fac valu. A Spcial Exampl: A consol bond or prptuity. Such a bond has a fixd coupon, $C, that is paid indfinitly into th futur; th principal is nvr rdmd. $ PV $C $C 1 i ( 1 ) c i c L L or $C $ PV i c 4) - Discount Bond is a dbt instrumnt that is issud at a discount and pays th fac valu at maturity. In othr words, th fac valu rprsnts th total of principal and intrst. $FV $PD i PD discount pric $PD Exampl: Considr a discountd bond sold at a pric of $975 today with a fac valu of to b paid at maturity. $975 or i or.56% $975 or aftr adjusting for annual yilds: $975 i * 360/days to maturity (say 90) 10.6% $975

3 Holding priod yild th distinction btwn intrst rats and rturns. Th pric of most forms of dbt fluctuats ovr tim whil th yild to maturity can b valuatd with crtainty. Thrfor, an individual may not want to hold a dbt until maturity, particularly whn a markt xists in which it can b tradd and whn individuals gambl about th futur cours of intrst rats. Capital Gains or Losss: changs in th markt valu of dbt instrumnt btwn th tim thy ar purchasd and th tim thy ar sold. Thus, holding priod yild th rturn for th priod of ownrship may b mor rlvant than th yild to maturity ithr bcaus a financial instrumnt is not hld until maturity or bcaus it was not purchasd at th tim of issu. RET C PV t PVt PV t or RET i C t g Rat of capital gain (or loss) Exampl 1: Suppos that today s intrst rat is 10% and an invstor purchass at fac valu a bond (Coupon bond) that maturs 0 yars from today. Th yild to maturity is 10%. Now suppos that th intrst rat riss to 15% som tim aftr th first yar and rmains at that lvl until th bond maturs. What is th ffct of th intrst rat chang on th prsnt valu or pric of th bond? If th bond has 19 yars lft to maturity, its pric is found by solving: PV L $ t 0 19 ( ) ( ) ( 1.15) $

4 Capital loss du to an incras in th intrst rat is: $ $ or 30.9% Total rturn 30.9% 10% 0.9% If, howvr, th bond has only on yar lft to maturity, its pric bcoms: $ PV L ( 1.10) ( 1.10) ( 1.15) t $91.47 Capital loss du to an incras in th intrst rat is: $91.47 $ or 8.75% Total rturn 8.75% 10% 1.5% Thus, th longr is th tim to maturity th largr th ffct on th pric of th bond (du to a chang in th intrst rat), and thus on its rturn. Exampl : On yar rturns whn intrst rat changs. Suppos a 10% coupon bond with a fac valu of is slling at par with yars to maturity. Calculat th rturn if th intrst rat ris from 10% to 0%? Th initial pric of th coupon bond is PV t, sinc th intrst rat quals th coupon rat whn th bond is at par. Nxt yar, whn th bond has on yar to maturity, th pric is: 4

5 PV (1 0.0) (1 0.) t 1 $917. Th on yar capital loss is givn as: $917 $ or 8.3% Th on-yar rturn 8.3% 10% 1.7% Rdo th xampl if th bond has 5 yars to maturity: Th initial pric of th coupon bond is still PV t, sinc th intrst rat quals th coupon rat whn th bond is at par. Nxt yar, whn th bond has four yars to maturity, th pric is: PV (1 0.0) (1 0.0) (1 0.0) (1 0.0) (1 0.) t $741. Th on yar capital loss is givn as: $741 $ or 5.9% Th on-yar rturn 5.9% 10% 15.9% Thus, whn intrst rats chang, a bond with a longr trm to maturity has a largr chang in its pric and hnc mor intrst rat risk than a bond with a shortr trm to maturity. 5

6 Not: Th fact that two bonds hav th sam trm to maturity dos not man that thy hav th sam intrst-rat risk. For instanc, a long-trm discount bond with 10 yars to maturity, a so-calld zrocoupon bond, maks all of its paymnts at th nd of th 10 yars, whras a 10% coupon bond with 10 yars to maturity maks cash paymnts bfor th maturity dat. Sinc th coupon bond maks paymnts arlir than th zro-coupon bond, w might intuitivly guss that th coupon bond s ffctiv maturity, th trm to maturity that accuratly masurs intrst-rat risk, is shortr than it is for th zrocoupon discount bond. : Rat of Capital Gain Exampl 3: Calculat th rat of capital gain or loss on a 10-yar zro-coupon bond (discount bond) for which th intrst rat has incrasd from 10% to 0%. Th bond has a fac valu of $1,000. Th pric of th bond today $ (1 0.1) Th pric of th bond on-yar from now $ (1 0.) $ $ Th rat of capital gain (or loss) $ or 49.7% Th on-yar rturn 49.7% 10% 39.7% But, as alrady calculatd in Tabl 4- in Chaptr 4, th capital loss on th 10% tn-yar coupon bond is -40.3%. W s that intrst-rat risk for th tn-yar coupon bond is lss than for th tn-yar zro-coupon bond (discount bond), so th 6

7 ffctiv maturity on th coupon bond (which masurs intrst-rat risk) is, as xpctd, shortr than th ffctiv maturity on th zro-coupon bond. Maturity and Volatility of Bond Rturns Only bonds whos rturns yilds is on with maturity holding priod. For bonds with maturity > holding priod, i P B capital loss. Longr is maturity, gratr is pric chang associatd with intrst rat chang. Longr is maturity, mor rturn changs with chang in intrst rat. Bond with high initial intrst rat can still hav ngativ rturn if i. Bottom lin: Prics (of bonds) and rturns mor volatil for long-trm bonds bcaus thy hav highr intrst rat risk. Distinction btwn Ral and Nominal Intrst Rats So far w hav considrd intrst rats and yilds without any considration of th ffcts of inflation or of taxs. Th Rol of Expctd Inflation Th intrst rat that is adjustd for xpctd changs in th pric lvl (inflation) so that it mor accuratly rflcts th tru cost of borrowing is calld th ral intrst rat. or i r π r i π --- Fishr Equation 7

8 Ral intrst calculations that us π instad of π yild th xpost ral intrst rat whil computation of r that us som proxy for π rsult in th xant ral intrst rat. It is asy to vrify that whn th ral intrst rat is low, thr ar gratr incntivs to borrow and fwr incntivs to lnd. Taxs Th ral intrst rat calculations implicit in th Fishr quation ar approximat bcaus ral intrst incom in Canada is subjct to incom tax. If τ is th incom tax rat, thn th aftr-tax ral intrst rat can b calculatd at as: r i( 1-τ ) π Exampl: If i 5% and π %, thn xant r i π 5% % 3%. Suppos, τ 30%, thn xant aftr-tax ral intrst rat would b: ( 1-τ ) 0.05( 1 0.3) % r i π 8